Let T₁,...,Tₙ be bounded linear operators on a complex Hilbert space H. Then there are compact operators K₁,...,Kₙ ∈ B(H) such that the closure of the joint numerical range of the n-tuple (T₁-K₁,...,Tₙ-Kₙ) equals the joint essential numerical range of (T₁,...,Tₙ). This generalizes the corresponding result for n = 1. We also show that if S ∈ B(H) and n ∈ ℕ then there exists a compact operator K ∈ B(H) such that $\left|\right|(S-K)ⁿ\left|\right|={\left|\right|Sⁿ\left|\right|}_e$. This generalizes results of C. L. Olsen.

Let W(A) and $W_e\left(A\right)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that $W_e\left(A\right)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, $W_e\left(A\right)$ can be obtained as the intersection of all sets of the form $cl\left(W(A₁,...,A_{i+1},A_i+F,A_{i+1},...,Aₘ)\right)$, where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in $W_e\left(A\right)$ as star centers....

We study the numerical radius of Lipschitz operators on Banach spaces. We give its basic properties. Our main result is a characterization of finite-dimensional real Banach spaces with Lipschitz numerical index 1. We also explicitly compute the Lipschitz numerical index of some classical Banach spaces.